#6629190

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628867
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60359	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,25)(13,20)(14,19)(15,18)(17,26)(21,35)(22,32)(23,31)(24,30)(27,34)(28,33)(29,36)(37,40)(38,39);(2,5)(6,8)(9,11,15,24)(10,16,18,30)(12,21,27,37)(13,22,14,23)(17,29,33,39)(19,31,20,32)(25,35,34,40)(26,36,28,38);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,28)(14,26)(15,27)(16,29)(18,33)(19,34)(20,25)(22,38)(23,36)(24,37)(30,39)(31,40)(32,35);(9,12)(10,17)(11,21)(13,26)(14,28)(15,27)(16,29)(18,33)(19,25)(20,34)(22,36)(23,38)(24,37)(30,39)(31,35)(32,40);(10,18)(16,30)(17,33)(19,20)(25,34)(29,39)(31,32)(35,40)

Polycyclic presentation:

2^9|b^2=i,c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,h^a=hi,g^c=gi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536386);